A windhover is a name for a kestrel, which is a small, common falcon. It is known for hovering in the air (wind hover) while facing the wind. Hopkins uses "Falcon" instead of windhover or kestrel because Falcon had/has a more regal sense. He also uses the word "dauphin" (son of a king) to emphasize how grand and royal the falcon is; this is also a reference to Christ. The falcon "rebuffs" (rejects or refuses) the "big wind." The bird uses its skill in flying/hovering to remain steady even when a big wind swoops in and nearly breaks its steady flight. The speaker admires this: "the mastery of the thing!"
The speaker continues with praises (brute beauty, valor). A fire "breaks" from the bird like a glow of light. Consider this line with the dedication of the poem: "To Christ our Lord." Christ is associated with light, the light of the world.
The speaker uses the word "buckle," which means to break but also to fasten. This suggests the falcon is vulnerable, about to be broken or pushed away by the wind, but also that the falcon is steadfast, buckled to his place in the sky. This is all a literal reading of the falcon but it is also an allegory for Christ and the crucifixion. Christ is a "master" of humanity and spiritual wisdom. In the end, he is broken (buckle) in the sky, hanging on the cross but his presence and message is steadfast and unbroken: buckled to the heavens, to truth, and to love. And he is then seen and/or associated with radiant light (the "fire" and "gash gold-vermilion"—gold light and the red vermilion blood). So, the falcon is like Christ, both images held in the air, both vulnerable but completely strong and masters of their respective domains, both silhouetted or surrounded by light.